[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83677-en":3,"doc-seo-83677-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83677,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Mitigating Numerical Stiffness in Least-Squares Formulations of Elliptic PDEs for Physics-Informed Neural Networks","The work provides theoretical analysis of H−1 residual loss formulations for physics-informed neural networks (PINNs) applied to learning solutions of elliptic partial differential equations (PDEs) with arbitrary, nonzero Dirichlet boundary conditions. Standard PINN losses use L2-based mean squared error (MSE) interior and boundary terms, whose imbalanced magnitudes can induce numerical stiffness, ill-conditioning, and slow convergence. The study analyzes H−1 discretizations that rebalance PDE loss, improve conditioning, and mitigate stiffness, validated by operator-level and PINN experiments for Poisson and stationary incompressible Navier–Stokes.","arXiv :2607 .02726v1 [math .NA] 2 Jul 2026  \nMitigating Numerical Stiffness in Least-Squares Formulations of Elliptic PDEs for Physics-Informed  \nNeural Networks  \nPhil-Alexander Hofmann∗, Michael Hecht†  \nJuly 7, 2026  \nAbstract  \nWe present theoretical insights into H −1 residual loss formulations of physics-informed neural networks (PINNs) for learning solutions of partial differential equations (PDEs) . Standard PINN formulations use a multi-term loss functional consisting of interior and boundary loss terms that are based on L2-residuals and discretized as mean square errors (MSE) . Imbalanced magnitudes of these terms cause numerical stiffness phenomena, resulting in ill-conditioning and slow convergence. In this work, we analyze discretizations of the H −1-norm that are used in the context of elliptic PDEs with arbitrary, nonzero Dirichlet boundary conditions.  \nWe prove that these H −1 discretizations rebalance the PDE loss, improve conditioning, and mitigate stiffness effects compared with the standard MSE discretization. We validate our theoretic results through operator-level experiments with randomly sampled residuals and PINN experiments for the Poisson and stationary incompressible Navier-Stokes equations.  \nThese experiments confirm the numerical effectiveness of the proposed rebalancing for elliptic PDEs and, more broadly, for problems with elliptic behavior.  \nKeywords Physics-informed neural networks · Negative Sobolev norm · Numerical stiffness · PDElearning  \n1 Introduction  \nAcross disciplines, partial differential equations (PDEs) are ubiquitous, serving as the primary framework for expressing physical laws and dynamics in systems such as fluid flow, heat transfer, and electromagnetism [12, 22] . Although, classic numerical methods for PDEs have been developed into a rich and rigorous theory, including finite element methods [16], finite difference methods [26], finite volume methods [17], spectral methods [6, 14], and meshfree methods [27] . However, their applicability is typically tied to inherent choices of meshes, basis functions, and boundary constraints. This inflexibility  \n∗ Center for Advanced Systems Understanding, Helmholtz-Zentrum Dresden-Rossendorf e.V., Untermarkt 20, 02826 Görlitz, Germany, [p.hofmann@hzdr.de](p.hofmann@hzdr.de)  \n†Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, [michael.hecht@math.uni.wroc.pl](michael.hecht@math.uni.wroc.pl)  \nhas long been recognized and addressed within Rayleigh–Ritz–Galerkin methods [4, 5 , 11] . Here, trial spaces that do not need to satisfy boundary conditions a priori and rely on boundary constraints enforced weakly or, alternatively, by boundary penalty terms.  \nSince 2017, physics–informed neural networks (PINNs) have emerged as a framework for solving ordinary and partial differential equations through residual-based losses [33–35] . Despite their empirical success across a wide range of applications [13, 25 , 38], it is well-known that PINNs suffer from severe stiffness and training pathologies [21, 30, 40] .  \nMotivated by the need to mitigate stiffness caused by boundary penalties and gradient imbalance, the H −1 residual loss formulation exploits the connection between PINNs and Rayleigh-Ritz-Galerkin methods [37] . The existing analysis in [37], however, was restricted to surrogate models satisfying homogeneous boundary conditions and therefore did not cover the full originally proposed objective.  \nHere, we complete the study of stiffness phenomena by extending the analysis to surrogate models with arbitrary nonzero Dirichlet boundary values. In particular, using an estimate of Lions and Magenes [28], we combine an H −1 interior residual loss with an H 1/2 boundary residual loss, yielding a formulation that is continuous in the H 1 norm.  \n1.1 Contribution  \nClassically, a physics-informed neural network (PINN) is a neural network  \nuθ : Ω ⊂ Rm → R (1)  \nparameterized by θ ∈ Rd , which is used to ","cbCaikPc8YNtStcO","https://ap.wps.com/l/cbCaikPc8YNtStcO","pdf",1772961,3,1,31,"English","en",105,"# Abstract\n# Introduction\n## Contribution","[{\"question\":\"Why do standard PINN losses suffer from numerical stiffness?\",\"answer\":\"Standard PINN losses combine interior and boundary terms discretized as mean squared errors (MSE). When their magnitudes are imbalanced, the PDE residual dominates, producing ill-conditioning and slow, unstable convergence with boundary artifacts.\"},{\"question\":\"What is the main idea of using H−1 residual loss for elliptic PDEs?\",\"answer\":\"The document analyzes H−1-norm discretizations for elliptic PDEs with arbitrary nonzero Dirichlet boundary conditions. These discretizations reweight the PDE loss relative to boundary terms, improving conditioning and mitigating stiffness effects.\"},{\"question\":\"How are the theoretical results validated?\",\"answer\":\"The authors validate the rebalancing and conditioning claims using operator-level experiments with randomly sampled residuals and PINN experiments for the Poisson equation and stationary incompressible Navier–Stokes equations.\"}]",1784189682,78,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"mitigating-numerical-stiffness-in-least-squares-formulations-of-elliptic-pdes-for-physics-informed-neural-networks","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/mitigating-numerical-stiffness-in-least-squares-formulations-of-elliptic-pdes-for-physics-informed-neural-networks/83677/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why do standard PINN losses suffer from numerical stiffness?","Question",{"text":75,"@type":76},"Standard PINN losses combine interior and boundary terms discretized as mean squared errors (MSE). When their magnitudes are imbalanced, the PDE residual dominates, producing ill-conditioning and slow, unstable convergence with boundary artifacts.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main idea of using H−1 residual loss for elliptic PDEs?",{"text":80,"@type":76},"The document analyzes H−1-norm discretizations for elliptic PDEs with arbitrary nonzero Dirichlet boundary conditions. These discretizations reweight the PDE loss relative to boundary terms, improving conditioning and mitigating stiffness effects.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the theoretical results validated?",{"text":84,"@type":76},"The authors validate the rebalancing and conditioning claims using operator-level experiments with randomly sampled residuals and PINN experiments for the Poisson equation and stationary incompressible Navier–Stokes equations.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]